Package org.apache.commons.math4.distribution

Class EmpiricalDistribution

java.lang.Object
org.apache.commons.math4.distribution.AbstractRealDistribution
org.apache.commons.math4.distribution.EmpiricalDistribution
All Implemented Interfaces:
java.io.Serializable, RealDistribution, ContinuousDistribution
public class EmpiricalDistribution
extends AbstractRealDistribution
implements ContinuousDistribution

Represents an empirical probability distribution -- a probability distribution derived from observed data without making any assumptions about the functional form of the population distribution that the data come from.

An EmpiricalDistribution maintains data structures, called distribution digests, that describe empirical distributions and support the following operations:

  • loading the distribution from a file of observed data values
  • dividing the input data into "bin ranges" and reporting bin frequency counts (data for histogram)
  • reporting univariate statistics describing the full set of data values as well as the observations within each bin
  • generating random values from the distribution
Applications can use EmpiricalDistribution to build grouped frequency histograms representing the input data or to generate random values "like" those in the input file -- i.e., the values generated will follow the distribution of the values in the file.

The implementation uses what amounts to the Variable Kernel Method with Gaussian smoothing:

Digesting the input file

  1. Pass the file once to compute min and max.
  2. Divide the range from min-max into binCount "bins."
  3. Pass the data file again, computing bin counts and univariate statistics (mean, std dev.) for each of the bins
  4. Divide the interval (0,1) into subintervals associated with the bins, with the length of a bin's subinterval proportional to its count.
Generating random values from the distribution
  1. Generate a uniformly distributed value in (0,1)
  2. Select the subinterval to which the value belongs.
  3. Generate a random Gaussian value with mean = mean of the associated bin and std dev = std dev of associated bin.

EmpiricalDistribution implements the RealDistribution interface as follows. Given x within the range of values in the dataset, let B be the bin containing x and let K be the within-bin kernel for B. Let P(B-) be the sum of the probabilities of the bins below B and let K(B) be the mass of B under K (i.e., the integral of the kernel density over B). Then set P(X < x) = P(B-) + P(B) * K(x) / K(B) where K(x) is the kernel distribution evaluated at x. This results in a cdf that matches the grouped frequency distribution at the bin endpoints and interpolates within bins using within-bin kernels.

USAGE NOTES:
  • The binCount is set by default to 1000. A good rule of thumb is to set the bin count to approximately the length of the input file divided by 10.
  • The input file must be a plain text file containing one valid numeric entry per line.
See Also:
Serialized Form

  • Field Details

  • Constructor Details

    • EmpiricalDistribution

      public EmpiricalDistribution()
      Creates a new EmpiricalDistribution with the default bin count.

    • EmpiricalDistribution

      public EmpiricalDistribution​(int binCount)
      Creates a new EmpiricalDistribution with the specified bin count.
      Parameters:
      binCount - number of bins. Must be strictly positive.
      Throws:
      NotStrictlyPositiveException - if binCount <= 0.

  • Method Details

    • load

      public void load​(double[] in) throws NullArgumentException
      Computes the empirical distribution from the provided array of numbers.
      Parameters:
      in - the input data array
      Throws:
      NullArgumentException - if in is null

    • load

      public void load​(java.net.URL url) throws java.io.IOException, NullArgumentException, ZeroException
      Computes the empirical distribution using data read from a URL.

      The input file must be an ASCII text file containing one valid numeric entry per line.

      Parameters:
      url - url of the input file
      Throws:
      java.io.IOException - if an IO error occurs
      NullArgumentException - if url is null
      ZeroException - if URL contains no data

    • load

      public void load​(java.io.File file) throws java.io.IOException, NullArgumentException
      Computes the empirical distribution from the input file.

      The input file must be an ASCII text file containing one valid numeric entry per line.

      Parameters:
      file - the input file
      Throws:
      java.io.IOException - if an IO error occurs
      NullArgumentException - if file is null

    • getSampleStats

      public StatisticalSummary getSampleStats()
      Returns a StatisticalSummary describing this distribution. Preconditions:
      • the distribution must be loaded before invoking this method
      Returns:
      the sample statistics
      Throws:
      java.lang.IllegalStateException - if the distribution has not been loaded

    • getBinCount

      public int getBinCount()
      Returns the number of bins.
      Returns:
      the number of bins.

    • getBinStats

      public java.util.List<SummaryStatistics> getBinStats()
      Returns a List of SummaryStatistics instances containing statistics describing the values in each of the bins. The list is indexed on the bin number.
      Returns:
      List of bin statistics.

    • getUpperBounds

      public double[] getUpperBounds()

      Returns a fresh copy of the array of upper bounds for the bins. Bins are:
      [min,upperBounds[0]],(upperBounds[0],upperBounds[1]],..., (upperBounds[binCount-2], upperBounds[binCount-1] = max].

      Note: In versions 1.0-2.0 of commons-math, this method incorrectly returned the array of probability generator upper bounds now returned by getGeneratorUpperBounds().

      Returns:
      array of bin upper bounds
      Since:
      2.1

    • getGeneratorUpperBounds

      public double[] getGeneratorUpperBounds()

      Returns a fresh copy of the array of upper bounds of the subintervals of [0,1] used in generating data from the empirical distribution. Subintervals correspond to bins with lengths proportional to bin counts.

      Preconditions:
      • the distribution must be loaded before invoking this method

      In versions 1.0-2.0 of commons-math, this array was (incorrectly) returned by getUpperBounds().

      Returns:
      array of upper bounds of subintervals used in data generation
      Throws:
      java.lang.NullPointerException - unless a load method has been called beforehand.
      Since:
      2.1

    • isLoaded

      public boolean isLoaded()
      Property indicating whether or not the distribution has been loaded.
      Returns:
      true if the distribution has been loaded

    • probability

      public double probability​(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X = x). In other words, this method represents the probability mass function (PMF) for the distribution.
      Specified by:
      probability in interface ContinuousDistribution
      Overrides:
      probability in class AbstractRealDistribution
      Parameters:
      x - Point at which the PMF is evaluated.
      Returns:
      zero.
      Since:
      3.1

    • density

      public double density​(double x)
      Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

      Returns the kernel density normalized so that its integral over each bin equals the bin mass.

      Algorithm description:

      1. Find the bin B that x belongs to.
      2. Compute K(B) = the mass of B with respect to the within-bin kernel (i.e., the integral of the kernel density over B).
      3. Return k(x) * P(B) / K(B), where k is the within-bin kernel density and P(B) is the mass of B.
      Specified by:
      density in interface ContinuousDistribution
      Parameters:
      x - Point at which the PDF is evaluated.
      Returns:
      the value of the probability density function at x.
      Since:
      3.1

    • cumulativeProbability

      public double cumulativeProbability​(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

      Algorithm description:

      1. Find the bin B that x belongs to.
      2. Compute P(B) = the mass of B and P(B-) = the combined mass of the bins below B.
      3. Compute K(B) = the probability mass of B with respect to the within-bin kernel and K(B-) = the kernel distribution evaluated at the lower endpoint of B
      4. Return P(B-) + P(B) * [K(x) - K(B-)] / K(B) where K(x) is the within-bin kernel distribution function evaluated at x.
      If K is a constant distribution, we return P(B-) + P(B) (counting the full mass of B).
      Specified by:
      cumulativeProbability in interface ContinuousDistribution
      Parameters:
      x - Point at which the CDF is evaluated.
      Returns:
      the probability that a random variable with this distribution takes a value less than or equal to x.
      Since:
      3.1

    • inverseCumulativeProbability

      public double inverseCumulativeProbability​(double p) throws OutOfRangeException
      Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is
      • inf{x in R | P(X<=x) >= p} for 0 < p <= 1,
      • inf{x in R | P(X<=x) > 0} for p = 0.
      The default implementation returns

      Algorithm description:

      1. Find the smallest i such that the sum of the masses of the bins through i is at least p.
      2. Let K be the within-bin kernel distribution for bin i.
        Let K(B) be the mass of B under K.
        Let K(B-) be K evaluated at the lower endpoint of B (the combined mass of the bins below B under K).
        Let P(B) be the probability of bin i.
        Let P(B-) be the sum of the bin masses below bin i.
        Let pCrit = p - P(B-)
      3. Return the inverse of K evaluated at
        K(B-) + pCrit * K(B) / P(B)
      Specified by:
      inverseCumulativeProbability in interface ContinuousDistribution
      Overrides:
      inverseCumulativeProbability in class AbstractRealDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
      Throws:
      OutOfRangeException
      Since:
      3.1

    • getMean

      public double getMean()
      Gets the mean of this distribution.
      Specified by:
      getMean in interface ContinuousDistribution
      Returns:
      the mean, or Double.NaN if it is not defined.
      Since:
      3.1

    • getVariance

      public double getVariance()
      Gets the variance of this distribution.
      Specified by:
      getVariance in interface ContinuousDistribution
      Returns:
      the variance, or Double.NaN if it is not defined.
      Since:
      3.1

    • getSupportLowerBound

      public double getSupportLowerBound()
      Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. inf {x in R | P(X <= x) > 0}.
      Specified by:
      getSupportLowerBound in interface ContinuousDistribution
      Returns:
      the lower bound of the support.
      Since:
      3.1

    • getSupportUpperBound

      public double getSupportUpperBound()
      Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. inf {x in R | P(X <= x) = 1}.
      Specified by:
      getSupportUpperBound in interface ContinuousDistribution
      Returns:
      the upper bound of the support.
      Since:
      3.1

    • isSupportConnected

      public boolean isSupportConnected()
      Indicates whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.
      Specified by:
      isSupportConnected in interface ContinuousDistribution
      Returns:
      whether the support is connected.
      Since:
      3.1

    • createSampler

      public ContinuousDistribution.Sampler createSampler​(UniformRandomProvider rng)
      Creates a sampler.
      Specified by:
      createSampler in interface ContinuousDistribution
      Overrides:
      createSampler in class AbstractRealDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.

    • getKernel

      protected ContinuousDistribution getKernel​(SummaryStatistics bStats)
      The within-bin smoothing kernel. Returns a Gaussian distribution parameterized by bStats, unless the bin contains only one observation, in which case a constant distribution is returned.
      Parameters:
      bStats - summary statistics for the bin
      Returns:
      within-bin kernel parameterized by bStats